The simplified expression for TDF has also been widely used to estimate the twist-angle dependence of conductivity from the Fermi velocity as \(\sigma (\theta ) \approx \sigma (0)[v_f(\theta )/v_f]^2\)  [1]} at low temperatures and carrier concentrations. However, the TDF also includes the temperature-dependent Fermi window (\(\partial f_0/\partial E\) ) whose width is \(\approx 5 k_B T\)  [2]} so transport occurs over a range of energies around \(E_f\) , which widens the peak in S\(_{eff}\)  and the minimum in \(\sigma _eff\)  as functions of twist angle. This trend is illustrated by observing the S\(_{eff}\)  and the corresponding \(\sigma _{eff}\)  for different twist angles in REF (e). Taken together, they produce a peak in PF at room temperature at twist angles around \(\theta \)  = 1.3\(^\circ \)  for both electrons and holes, with maximum PF reaching around 40 mWm\(^{-1}\)  K\(^{-2}\)  (Fig. REF (d)). The peaks in PF occur at a carrier density of 3.9 \(\times \)  10\(^{12}\)  cm\(^{-2}\)  in holes and 4.7 \(\times \)  10\(^{12}\)  cm\(^{-2}\)  in electrons, near complete band-filling \(n_s = \pm \)  4.1 \(\times \)  10\(^{12}\)  cm\(^{-2}\) . This stands in contrast to other notable properties of TBG, superconductivity and Mott insulating behavior, which typically occur at half-filling. The range of twist angles we simulated domain is limited to angles above 0.65\(^\circ \)  due to limitations of the continuum model that we used for band structure calculations. This limitation comes from the inclusion of atomic relaxation in TBG that suppress the secondary magic-angle below \(\theta \)  = 0.65\(^\circ \) . However, the band gaps occuring at magic angles below 0.65\(^\circ \)  are smaller [3]}, so we expect them to result in lesser enhancements of Seebeck and PF.
